# Worst-Case Optimal Join Algorithms: Techniques, Results, and Open Problems

## TL;DR

This paper introduces worst-case optimal join algorithms, highlighting their theoretical foundations, practical implementations, and open research problems, emphasizing their significance across multiple disciplines and real-world data systems.

## Contribution

It provides a comprehensive overview of the design, analysis, and key techniques of worst-case optimal join algorithms, connecting theory with practical applications.

## Key findings

- Algorithms match the worst-case output size bounds.
- Connections between join algorithms and information theory.
- Identification of fundamental open problems in the field.

## Abstract

Worst-case optimal join algorithms are the class of join algorithms whose runtime match the worst-case output size of a given join query. While the first provably worst-case optimal join algorithm was discovered relatively recently, the techniques and results surrounding these algorithms grow out of decades of research from a wide range of areas, intimately connecting graph theory, algorithms, information theory, constraint satisfaction, database theory, and geometric inequalities. These ideas are not just paperware: in addition to academic project implementations, two variations of such algorithms are the work-horse join algorithms of commercial database and data analytics engines. This paper aims to be a brief introduction to the design and analysis of worst-case optimal join algorithms. We discuss the key techniques for proving runtime and output size bounds. We particularly focus on the fascinating connection between join algorithms and information theoretic inequalities, and the idea of how one can turn a proof into an algorithm. Finally, we conclude with a representative list of fundamental open problems in this area.